Theorem txindis 23555

Description: The topological product of indiscrete spaces is indiscrete. (Contributed by Mario Carneiro, 14-Aug-2015.)

Assertion

Ref	Expression
txindis	⊢ ({∅, 𝐴} ×t {∅, 𝐵}) = {∅, (𝐴 × 𝐵)}

Proof of Theorem txindis

Dummy variables 𝑥 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		neq0 4311	. . . . . . 7 ⊢ (¬ 𝑥 = ∅ ↔ ∃𝑦 𝑦 ∈ 𝑥)
2		indistop 22923	. . . . . . . . . . 11 ⊢ {∅, 𝐴} ∈ Top
3		indistop 22923	. . . . . . . . . . 11 ⊢ {∅, 𝐵} ∈ Top
4		eltx 23489	. . . . . . . . . . 11 ⊢ (({∅, 𝐴} ∈ Top ∧ {∅, 𝐵} ∈ Top) → (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) ↔ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ {∅, 𝐴}∃𝑤 ∈ {∅, 𝐵} (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)))
5	2, 3, 4	mp2an 692	. . . . . . . . . 10 ⊢ (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) ↔ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ {∅, 𝐴}∃𝑤 ∈ {∅, 𝐵} (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥))
6		rsp 3223	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝑥 ∃𝑧 ∈ {∅, 𝐴}∃𝑤 ∈ {∅, 𝐵} (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥) → (𝑦 ∈ 𝑥 → ∃𝑧 ∈ {∅, 𝐴}∃𝑤 ∈ {∅, 𝐵} (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)))
7	5, 6	sylbi 217	. . . . . . . . 9 ⊢ (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) → (𝑦 ∈ 𝑥 → ∃𝑧 ∈ {∅, 𝐴}∃𝑤 ∈ {∅, 𝐵} (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)))
8		elssuni 4897	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) → 𝑥 ⊆ ∪ ({∅, 𝐴} ×t {∅, 𝐵}))
9		indisuni 22924	. . . . . . . . . . . . . . 15 ⊢ ( I ‘𝐴) = ∪ {∅, 𝐴}
10		indisuni 22924	. . . . . . . . . . . . . . 15 ⊢ ( I ‘𝐵) = ∪ {∅, 𝐵}
11	2, 3, 9, 10	txunii 23514	. . . . . . . . . . . . . 14 ⊢ (( I ‘𝐴) × ( I ‘𝐵)) = ∪ ({∅, 𝐴} ×t {∅, 𝐵})
12	8, 11	sseqtrrdi 3985	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) → 𝑥 ⊆ (( I ‘𝐴) × ( I ‘𝐵)))
13	12	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) ∧ (𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵})) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → 𝑥 ⊆ (( I ‘𝐴) × ( I ‘𝐵)))
14		ne0i 4300	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (𝑧 × 𝑤) → (𝑧 × 𝑤) ≠ ∅)
15	14	ad2antrl 728	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → (𝑧 × 𝑤) ≠ ∅)
16		xpnz 6120	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 ≠ ∅ ∧ 𝑤 ≠ ∅) ↔ (𝑧 × 𝑤) ≠ ∅)
17	15, 16	sylibr 234	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → (𝑧 ≠ ∅ ∧ 𝑤 ≠ ∅))
18	17	simpld 494	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → 𝑧 ≠ ∅)
19	18	neneqd 2930	. . . . . . . . . . . . . . . 16 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → ¬ 𝑧 = ∅)
20		simpll 766	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → 𝑧 ∈ {∅, 𝐴})
21		indislem 22921	. . . . . . . . . . . . . . . . . . 19 ⊢ {∅, ( I ‘𝐴)} = {∅, 𝐴}
22	20, 21	eleqtrrdi 2839	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → 𝑧 ∈ {∅, ( I ‘𝐴)})
23		elpri 4609	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ {∅, ( I ‘𝐴)} → (𝑧 = ∅ ∨ 𝑧 = ( I ‘𝐴)))
24	22, 23	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → (𝑧 = ∅ ∨ 𝑧 = ( I ‘𝐴)))
25	24	ord 864	. . . . . . . . . . . . . . . 16 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → (¬ 𝑧 = ∅ → 𝑧 = ( I ‘𝐴)))
26	19, 25	mpd 15	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → 𝑧 = ( I ‘𝐴))
27	17	simprd 495	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → 𝑤 ≠ ∅)
28	27	neneqd 2930	. . . . . . . . . . . . . . . 16 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → ¬ 𝑤 = ∅)
29		simplr 768	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → 𝑤 ∈ {∅, 𝐵})
30		indislem 22921	. . . . . . . . . . . . . . . . . . 19 ⊢ {∅, ( I ‘𝐵)} = {∅, 𝐵}
31	29, 30	eleqtrrdi 2839	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → 𝑤 ∈ {∅, ( I ‘𝐵)})
32		elpri 4609	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ {∅, ( I ‘𝐵)} → (𝑤 = ∅ ∨ 𝑤 = ( I ‘𝐵)))
33	31, 32	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → (𝑤 = ∅ ∨ 𝑤 = ( I ‘𝐵)))
34	33	ord 864	. . . . . . . . . . . . . . . 16 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → (¬ 𝑤 = ∅ → 𝑤 = ( I ‘𝐵)))
35	28, 34	mpd 15	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → 𝑤 = ( I ‘𝐵))
36	26, 35	xpeq12d 5662	. . . . . . . . . . . . . 14 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → (𝑧 × 𝑤) = (( I ‘𝐴) × ( I ‘𝐵)))
37		simprr 772	. . . . . . . . . . . . . 14 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → (𝑧 × 𝑤) ⊆ 𝑥)
38	36, 37	eqsstrrd 3979	. . . . . . . . . . . . 13 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → (( I ‘𝐴) × ( I ‘𝐵)) ⊆ 𝑥)
39	38	adantll 714	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) ∧ (𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵})) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → (( I ‘𝐴) × ( I ‘𝐵)) ⊆ 𝑥)
40	13, 39	eqssd 3961	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) ∧ (𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵})) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → 𝑥 = (( I ‘𝐴) × ( I ‘𝐵)))
41	40	ex 412	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) ∧ (𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵})) → ((𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥) → 𝑥 = (( I ‘𝐴) × ( I ‘𝐵))))
42	41	rexlimdvva 3192	. . . . . . . . 9 ⊢ (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) → (∃𝑧 ∈ {∅, 𝐴}∃𝑤 ∈ {∅, 𝐵} (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥) → 𝑥 = (( I ‘𝐴) × ( I ‘𝐵))))
43	7, 42	syld 47	. . . . . . . 8 ⊢ (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) → (𝑦 ∈ 𝑥 → 𝑥 = (( I ‘𝐴) × ( I ‘𝐵))))
44	43	exlimdv 1933	. . . . . . 7 ⊢ (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) → (∃𝑦 𝑦 ∈ 𝑥 → 𝑥 = (( I ‘𝐴) × ( I ‘𝐵))))
45	1, 44	biimtrid 242	. . . . . 6 ⊢ (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) → (¬ 𝑥 = ∅ → 𝑥 = (( I ‘𝐴) × ( I ‘𝐵))))
46	45	orrd 863	. . . . 5 ⊢ (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) → (𝑥 = ∅ ∨ 𝑥 = (( I ‘𝐴) × ( I ‘𝐵))))
47		vex 3448	. . . . . 6 ⊢ 𝑥 ∈ V
48	47	elpr 4610	. . . . 5 ⊢ (𝑥 ∈ {∅, (( I ‘𝐴) × ( I ‘𝐵))} ↔ (𝑥 = ∅ ∨ 𝑥 = (( I ‘𝐴) × ( I ‘𝐵))))
49	46, 48	sylibr 234	. . . 4 ⊢ (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) → 𝑥 ∈ {∅, (( I ‘𝐴) × ( I ‘𝐵))})
50	49	ssriv 3947	. . 3 ⊢ ({∅, 𝐴} ×t {∅, 𝐵}) ⊆ {∅, (( I ‘𝐴) × ( I ‘𝐵))}
51	9	toptopon 22838	. . . . . . 7 ⊢ ({∅, 𝐴} ∈ Top ↔ {∅, 𝐴} ∈ (TopOn‘( I ‘𝐴)))
52	2, 51	mpbi 230	. . . . . 6 ⊢ {∅, 𝐴} ∈ (TopOn‘( I ‘𝐴))
53	10	toptopon 22838	. . . . . . 7 ⊢ ({∅, 𝐵} ∈ Top ↔ {∅, 𝐵} ∈ (TopOn‘( I ‘𝐵)))
54	3, 53	mpbi 230	. . . . . 6 ⊢ {∅, 𝐵} ∈ (TopOn‘( I ‘𝐵))
55		txtopon 23512	. . . . . 6 ⊢ (({∅, 𝐴} ∈ (TopOn‘( I ‘𝐴)) ∧ {∅, 𝐵} ∈ (TopOn‘( I ‘𝐵))) → ({∅, 𝐴} ×t {∅, 𝐵}) ∈ (TopOn‘(( I ‘𝐴) × ( I ‘𝐵))))
56	52, 54, 55	mp2an 692	. . . . 5 ⊢ ({∅, 𝐴} ×t {∅, 𝐵}) ∈ (TopOn‘(( I ‘𝐴) × ( I ‘𝐵)))
57		topgele 22851	. . . . 5 ⊢ (({∅, 𝐴} ×t {∅, 𝐵}) ∈ (TopOn‘(( I ‘𝐴) × ( I ‘𝐵))) → ({∅, (( I ‘𝐴) × ( I ‘𝐵))} ⊆ ({∅, 𝐴} ×t {∅, 𝐵}) ∧ ({∅, 𝐴} ×t {∅, 𝐵}) ⊆ 𝒫 (( I ‘𝐴) × ( I ‘𝐵))))
58	56, 57	ax-mp 5	. . . 4 ⊢ ({∅, (( I ‘𝐴) × ( I ‘𝐵))} ⊆ ({∅, 𝐴} ×t {∅, 𝐵}) ∧ ({∅, 𝐴} ×t {∅, 𝐵}) ⊆ 𝒫 (( I ‘𝐴) × ( I ‘𝐵)))
59	58	simpli 483	. . 3 ⊢ {∅, (( I ‘𝐴) × ( I ‘𝐵))} ⊆ ({∅, 𝐴} ×t {∅, 𝐵})
60	50, 59	eqssi 3960	. 2 ⊢ ({∅, 𝐴} ×t {∅, 𝐵}) = {∅, (( I ‘𝐴) × ( I ‘𝐵))}
61		txindislem 23554	. . 3 ⊢ (( I ‘𝐴) × ( I ‘𝐵)) = ( I ‘(𝐴 × 𝐵))
62	61	preq2i 4697	. 2 ⊢ {∅, (( I ‘𝐴) × ( I ‘𝐵))} = {∅, ( I ‘(𝐴 × 𝐵))}
63		indislem 22921	. 2 ⊢ {∅, ( I ‘(𝐴 × 𝐵))} = {∅, (𝐴 × 𝐵)}
64	60, 62, 63	3eqtri 2756	1 ⊢ ({∅, 𝐴} ×t {∅, 𝐵}) = {∅, (𝐴 × 𝐵)}

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053   ⊆ wss 3911  ∅c0 4292  𝒫 cpw 4559  {cpr 4587  ∪ cuni 4867   I cid 5525   × cxp 5629  ‘cfv 6499  (class class class)co 7369  Topctop 22814  TopOnctopon 22831   ×_t ctx 23481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-1st 7947  df-2nd 7948  df-topgen 17383  df-top 22815  df-topon 22832  df-bases 22867  df-tx 23483

This theorem is referenced by: (None)